**Required reading:**

Moore and Reynolds, 298-316**
**Brindley and Brown, pages 411-436

**XRD Quantification of clay minerals
**Elemental analysis tells you what elements are in a
mixture... Qualitative XRD tells you where the elements are in a
mixture... Quantitative XRD and elemental analysis should agree.

The basic premise of quantitative XRD is that the intensity of a diffracted beam is proportional to amount of material present.

Remember! There are intrument, sample, and specimen sensitive factors can affect the intensity of a diffracted X-ray beam.

One of the major drawbacks of directly comparing peak
intensities to the amount of material present in the mixture has
to do with non-linear responses resulting from differences in
mass absorption of each phase in a mixture.

Specifically, when a mixture contains a weak absorber and strong absorber, the lines of the weaker component appears weaker and the strong component appears stronger in a non-linear fashion.

Recall from our previous lecture 3 , the mass absorption coefficient of a mixture is equal to the sums of the mass absorption coefficients of the components times their weight fraction:

where:

- µ* = mass absorption coefficent of sample
- û*= mass absorption coefficent of mixture
*I, J, K*... = components of a mixture- N = Number of components in mixture
- W = weight fraction of a component

**Primary and secondary extinction (the need for small
particle sizes):**

A perfect crystal normally exhibits a small amount of absorption. However, it is possible for additional attenuation of the beam in regions of Bragg reflection. This occurs in several ways such that the most intense lines are diminished. For the most intense line, this decrease can be up to a factor of 70 times (but not enough to make the line go totally extinct... hence a bit of a mis-nomer).

(1) The beam becomes increasingly weaker as it passes to deeper and deeper planes because those near the surface crystallites extract energy.

(2) Some wavlets interfere destructively as they undergo
secondary reflection from the underside of the atomic planes.
The diffracted beam is attenuated by back reflected wavelets.

These two effects result in a loss in beam intensity and is referred to as primary extinction. This effect varies as a with the angle of incidence, becoming less at higher angles where peak intensities are generally weaker.

Most crystals are composed of a mosaic of tiny imperfect
structures. In a powder there are a large number of "crystal
blocks" (i.e., coherent scattering domains) that are at the
correct Bragg angle for reflection. As a powder experiment is
conducted through out angular ranges, some of the crystallites
go in and some go out of the Bragg condition. What the
detector "sees" is the sum of successive contributions. The
slight mis-orentation of crystallites carries more energy than
if the sample was a perfect crystal. The powder condition
results in more absorption (i.e. increases effective
µ*). This effect is referred to as secondary extinction.

In other words... The total intensity of a diffracted beam is that which comes throughout the entire range of Bragg reflection. Slight mis-orientation of crystallites in a powder results in the additional contribution of diffracted wavelets. The crystallites near the surface, will reduce the intensity of diffracted wavelets from deeper within the crystal. This has an effect of a greater apparent µ* value, than that of a perfect crystal. Secondary extinction effect is greater for the stronger reflections.

These extinction effects can only be completely understood if
treated with dynamical theory. The bottom line here is that
powders with large particles (large crystallites > 5 micron)
will have biased intensities. The result is that the most
intense reflections will be reduced in intensity.

To minimize primary and secondary extinction, the particle size should be reduced to 5 - 10 µm.

See Table 5-9 (page 366 ) in Klug and Alexander, which shows
the variation in intensity measurements of the quartz peak at
3.34Å using different particle sizes. In essence, the table
illustrates that for 10 replicates of quartz intensity
measurements, the peak areas for particles ranging from 10 to 50
µm and particles ranging from 5 to 15 µm have lower means means
and higher percentage deviations (18.2% and 2.1%, respectively).
Whereas the <5 µm fraction has a greater intensities and
lower deviations.

It is assumed that the sample is a uniform mixture of N
components and there is no extinction of microabsorption (*i.e.*, the particle size is
sufficiently small).

The total intensity of the J^{th}
component of the mixture represented by some plane (*hkl*) = *i* is given by:

Where: *K*_{iJ} depends upon the nature
of component J and
instrumental parameters (*e.g.*, geometry).

*f*= the volume fraction of J in the mixture_{J}*r*= radius of diffractometer*e*and*m*= charge and mass of an electron*M*= Multiplicity factor- F = Structure factor for reflection i
- I
_{o}= Incident beam intensity - c = speed of light
- λ = wavelength
- θ = Bragg angle
*i, j, k,...*lines in diffraction pattern

It can be shown that the total intensity of the J^{th} component of
the mixture represented by some plane (*hkl*) = *i* can also be given by:

This equation is the underlying basis for all quantitative analysis by X-ray diffraction.

**Case 1:** Mixtures
where the *µ** of the unknown is the same as the
entire mixture. (not common except maybe calcite-aragonite or
quartz-cristobalite samples).

In this case, one can simply ratio the intensity of the i^{th} line in a
mixture to same line in the pure component.

Let (*I _{iJ}*)

Then:

Example: Mixture of quartz and cristobalite (polymorphs of SiO_{2}):

**Case 2:** Mixture of two components with
different (*i.e.*, µ** _{1}* not equal to µ*

For a pure phase:

In a mixture:

The theoretical curve for mixture of the two components is described by the equation:

The lines of the weaker component appears weaker and the strong component appears stronger

Mixtures of quartz (µ* = 34.4 cm^{2}/g) and KCl (µ* =
124 cm^{2}/g) or quartz and BeO (µ* = 7.9 cm^{2}/g)
using Cu radiation.

Think about geologic mixtures with gibbsite and goethite for
example....